242 ON THE COMPOSITION AND RESOLUTION OF STREAMS OF
whence,
cos2 (*2 - cO sin2 (ft - ft) + sin8 (a2 - a,) cos2 (ft + ft) = 0,
which leads to the very same conditions that have been already discussed in Art. 2. Hence two polarized streams coming from the same polarized source are capable of interfering perfectly if the polarizations are the same, not at all if the polarizations are opposite, and in intermediate cases of course in intermediate degrees.
5. When a stream of polarized light is resolved into two oppositely polarized streams, which are again compounded after their phases have been differently altered, we have from (9), taking account of (10) or (11),
1=^ + ^........................ (12),
so that the intensity of the resultant is equal to the sum of the intensities of the components, and is therefore constant, that is, independent of pl — p2, and is accordingly equal to what it was at first, when pl and p2 were each equal to zero, that is, equal to the intensity of the original stream.
It may be readily proved from the formulae (8) that it is only in the case in which the polarizations of the two components of the original polarized stream are opposite that the intensity of the original stream, whatever be the nature of its polarization, is equal to the sum of the intensities of the component streams. For, changing the sign of /\/(—1) in these formulae, multiplying the resulting equations, member for member, by the equations (8), and observing that if G' be what G becomes, GG' = ff* + A2 = c2, we find
(sin2 (ft - £ ) cos2 (a, - a ) + cos2 (ft + /3 ) sin8 (a2- a )} l<
(13).
= (sin2 (/3 — ft) cos2 (ojj — a ) -f cos2 (/? -f /3a) sin2 (c^— a ) j 1 c2 = (sin2 (ft - ft) cos2 (a2 - ax) + cos2 (ft + ft) sin2 (a,- a,)}"1 c 5
In order that itt (c2) may be equal to m (ct2) + 111 (c22), it is necessary that c2 be equal to c* -1- c22, because, whatever fluctuations cx and c2 may undergo in a moderate time, such as the tenth part of a second, c, and c2 are always proportional to c. Hence the sum of the quantities whose reciprocals are the coefficients of c,2 and c22 must be equal to that whose reciprocal is the coefficient of c2. Since this has to be true independently of